Singular intersections in split semiabelian varieties

Abstract

Let $\mathcal{G}$ be a split semiabelian variety of dimension $n\ge 2$ and let $\mathcal{C}\subset \mathcal{G}$ an irreducible curve, where everything is defined over $\overline{\mathbb{Q}}$. We prove that, under the natural hypothesis that $\mathcal{C}$ is not contained in any proper algebraic subgroup of $\mathcal{G}$, the set of points $P$ of $\mathcal{C}$ lying in an algebraic subgroup $H$ which intersects the curve tangentially at $P$ is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.

arXiv comment22 pages, comments are welcome!